Get Rich or Buy Trying - Part 1
Puzzle for the week of 10/6/24
Inspired by the Fiddler on the Proof (formerly The Riddler), X’s Puzzle Corner aims to produce a weekly puzzle for readers that enjoy math, probability, and algorithms. Please submit your solution! Solutions will be accepted until 11 pm the following Sunday after the puzzle is posted (in this case 10/13/24). While it isn’t required, I encourage you to opt to have your solution shared so that we all get the chance to see how others thought about and attempted the problem!
The answers of all those that volunteered their solutions will be posted around Wednesday at 10 am.
In many cases, I expect the readers will be better puzzlers than me so I make no guarantees the solutions are correct. I also make no promises about having worked out the solutions to the puzzles ahead of time so it may be the case that they’re very challenging. Part of the fun is finding out!
I hope you like gambling because this week’s puzzle is about….the stock market!
You decide you want to make your fortune off the stock market. So far you’ve had mixed results but today is his lucky day; you happen to be sitting next a prolific investor, we’ll call him Warren. Warren tells you that the secret to his success are two mystical oracles given to him by some old dead guy named Ben. What’s more, Warren is about to retire and is willing to give you these oracles if you can prove you’re worthy of wielding them.
The first oracle is able to tell with certainty whether a stock will go up or down that day. The second oracle is able to tell you with certainty whether a stock will go up that hour. The question you are presented with is; which oracle can get you better returns and by exactly how much?
Before tackling Warren’s question, let’s start with a warm up and some preliminaries.
We will consider two different types of behavior of this stock:
The stock has equal probability of going up or down each minute by p0/1000 where p0 is the initial value of the stock.
The stock has equal probability of going up to a value of 1.001*p_t-1 or down to a value .999*p_t-1 each minute (where p_t-1 is the value of the stock for the previous minute).
Your job for this first puzzle it to determine the distribution of the value of the stock after 1 day. Assume the stock is traded for 7 hours a day (a bit off, I know).
Good luck solving!
Please submit your answers here.
I'm a bit confused about the wording of the two behaviors. Is p0 the value at the start of the day, or the value at the start of each minute? If the former, the second behavior doesn't make sense, because it's saying the stock just jumps between those values. But if the latter, aren't the two behaviors identical?